Paired ofdm pilot symbols

ABSTRACT

Successive pairs of OFDM symbols are transmitted by an OFDM transmitter and received by an OFDM receiver. The successive pairs include a first pair of OFDM symbols. First and second OFDM symbols of the first pair both include pilot symbols on two subcarriers that are symmetric about a center carrier frequency. The two subcarriers are the same for the first and second OFDM symbols. The pilot symbols on the two subcarriers for the first and second OFDM symbols compose an orthogonal matrix. The OFDM receiver estimates frequency responses at frequencies including the frequencies of the two subcarriers and compensates for signal impairment based at least in part on the estimated frequency responses.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationsNo. 61/618,624, titled “Receiver-Side Estimation of and Compensation forSignal Impairments,” filed Mar. 30, 2012, and No. 61/719,326, titled“Receiver-Side Estimation of and Compensation for Signal Impairments,”filed Oct. 26, 2012, both of which are hereby incorporated by referencein their entirety.

TECHNICAL FIELD

The present embodiments relate generally to communication systems, andspecifically to communication systems that use pilot symbols tocompensate for signal impairments.

BACKGROUND OF RELATED ART

Transceivers are sensitive to various signal impairments that affect thequality of the transmitted and received signals. Signal impairments mayresult from non-idealities in the RF front-ends of the transceivers. Forexample, mismatched active and passive elements (e.g., quadraturemixers, filters, digital-to-analog converters, and/or analog-to-digitalconverters) in the I and Q (in-phase and quadrature) signal pathsintroduce I/Q mismatch impairments in transmitted and received signals.I/Q mismatch, which also may be referred to as I/Q offset, is present inboth the transmitter and receiver. In another example, carrier frequencyoffset in the receiver impairs received signals. Channel effects mayalso impair signals.

I/Q mismatch introduces an image signal that degrades signal quality.The signal-to-image ratio is typically around 25−30 dB, making I/Qmismatch an issue for systems targeting high spectral efficiency. I/Qmismatch is also frequency dependent, making I/Q mismatch an issue forwideband communication systems.

Accordingly, there is a need for techniques to estimate and compensatefor signal impairments.

BRIEF DESCRIPTION OF THE DRAWINGS

The present embodiments are illustrated by way of example and are notintended to be limited by the figures of the accompanying drawings. Likenumbers reference like elements throughout the drawings andspecification.

FIG. 1A illustrates a communications system in accordance with someembodiments.

FIG. 1B illustrates sources of signal impairment in the communicationssystem of FIG. 1A.

FIG. 2A is a block diagram of a direct-conversion transceiver inaccordance with some embodiments.

FIGS. 2B-2E are block diagrams illustrating signal impairments inaccordance with some embodiments.

FIG. 3A is a flowchart illustrating a method of estimating andcompensating for signal impairments in accordance with some embodiments.

FIG. 3B is a flowchart illustrating a two-phased method of estimatingand compensating for signal impairments in accordance with someembodiments.

FIGS. 4A-4D illustrate a periodic signal used for receiver-side I/Qmismatch estimation and compensation in accordance with someembodiments.

FIG. 5 is a block diagram of a communication device in accordance withsome embodiments.

FIG. 6 illustrates a channel matrix used for frequency-domain signalimpairment estimation and compensation in accordance with someembodiments.

FIG. 7 illustrates successive pairs of OFDM symbols with pilot symbolson different sets of subcarriers in accordance with some embodiments.

FIG. 8 is a flowchart showing a method of communicating between an OFDMtransmitter and an OFDM receiver in accordance with some embodiments.

DETAILED DESCRIPTION

Techniques are disclosed for transmitting and receiving pilot symbolsthat may be used to compensate for signal impairments.

In some embodiments, an orthogonal frequency-division multiplexing(OFDM) transmitter transmits successive pairs of OFDM symbols. Thesuccessive pairs include a first pair of OFDM symbols. First and secondOFDM symbols of the first pair both include pilot symbols on twosubcarriers that are symmetric about a center carrier frequency. The twosubcarriers are the same for the first and second OFDM symbols. Thepilot symbols on the two subcarriers for the first and second OFDMsymbols compose an orthogonal matrix.

In some embodiments, an OFDM receiver receives successive pairs of OFDMsymbols. The successive pairs include a first pair of OFDM symbols.First and second OFDM symbols of the first pair both include pilotsymbols on two subcarriers that are symmetric about a center carrierfrequency. The two subcarriers are the same for the first and secondOFDM symbols. The pilot symbols on the two subcarriers for the first andsecond OFDM symbols compose an orthogonal matrix. The OFDM receiverestimates frequency responses at frequencies including the frequenciesof the two subcarriers and compensates for signal impairment based atleast in part on the estimated frequency responses.

In the following description, numerous specific details are set forthsuch as examples of specific components, circuits, and processes toprovide a thorough understanding of the present disclosure. Also, in thefollowing description and for purposes of explanation, specificnomenclature is set forth to provide a thorough understanding of thepresent embodiments. However, it will be apparent to one skilled in theart that these specific details may not be required to practice thepresent embodiments. In other instances, well-known circuits and devicesare shown in block diagram form to avoid obscuring the presentdisclosure. The term “coupled” as used herein means connected directlyto or connected through one or more intervening components or circuits.Any of the signals provided over various buses described herein may betime-multiplexed with other signals and provided over one or more commonbuses. Additionally, the interconnection between circuit elements orsoftware blocks may be shown as buses or as single signal lines. Each ofthe buses may alternatively be a single signal line, and each of thesingle signal lines may alternatively be buses, and a single line or busmight represent any one or more of a myriad of physical or logicalmechanisms for communication between components. The present embodimentsare not to be construed as limited to specific examples described hereinbut rather to include within their scopes all embodiments defined by theappended claims.

FIG. 1A illustrates a communications system 100 in accordance with someembodiments. A transmitter 102 transmits a signal onto a channel 104,and a direct-conversion receiver 106 receives the signal from thechannel 104. In some embodiments, the channel 104 is wireless. In otherembodiments, the channel 104 is a wired link (e.g., a coaxial cable orother physical connection).

FIG. 1B illustrates sources of signal impairment, and thus signaldegradation, in the communications system 100 of FIG. 1A.Transmitter-side (Tx) I/Q mismatch 122 in the transmitter 102 causessignal impairment, as does receiver-side (Rx) I/Q mismatch 128 in thereceiver 106. The channel 104 introduces channel distortion 124, whichmay be linear distortion. Carrier frequency offset 126 in the receiver106, which results from the frequency of a local oscillator in thereceiver 106 differing from the frequency of a corresponding localoscillator in the transmitter 102, also causes signal impairment. Insome embodiments, channel distortion 124 includes multi-path effects andAdditive White Gaussian Noise (AWGN).

FIG. 2A is a block diagram of a direct-conversion transceiver 200 usingquadrature amplitude modulation (QAM) in accordance with someembodiments. The transceiver 200 may be included within a communicationdevice (e.g., communication device 500, FIG. 5), such as a wireless(e.g., WLAN) device or a device with a wired network connection. Asillustrated, the transceiver 200 includes a transmitter unit 210 and areceiver unit 250. The transmitter unit 210 of a first transceiver 200corresponds to the transmitter 102 (FIG. 1A) and the receiver unit 250of a second transceiver 200 corresponds to the receiver 106 (FIG. 1A),where the first transceiver 200 and second transceiver 200 are separatedby the channel 104 (FIG. 1A).

In some embodiments, the transmitter unit 210 includes an antenna 202, atransmitter analog front end (AFE) 220, and a transmitter basebandprocessor 240. The receiver unit 250 includes an antenna 201, a receiverAFE 260, and a receiver baseband processor 280. In some embodiments, thereceiver baseband processor 280 includes a signal impairmentcompensation unit 285 for estimating and compensating for signalimpairments introduced both in the transmitter (e.g., transmitter 102,FIG. 1A) and receiver (e.g., receiver 106, FIG. 1A).

In the example of FIG. 2A, the transmitter AFE 220 includes adigital-to-analog converter (DAC) 221A for the I signal path,amplifier/filter circuitry 222A for the I signal path, a localoscillator (LO) mixer 224A for the I signal path, a DAC 221B for the Qsignal path, amplifier/filter circuitry 222B for the Q signal path, anLO mixer 224B for the Q signal path, a combiner 272, a variable gainamplifier (VGA) 226, and a power amplifier (PA) 228. The mixers 224A and224B up-convert the I and Q signals from baseband directly to thecarrier frequency by mixing the I and Q signals with local oscillatorsignals LO(I) and LO(Q), where the frequency of the local oscillatorsignal is the carrier frequency. Mismatch between mixers 224A and 224B,between amplifiers/filters 222A and 222B, and/or between DACs 221A and221B results in transmitter-side I/Q mismatch. The combiner 272 combinesthe up-converted I and Q signals.

The receiver AFE 260 includes a low-noise amplifier (LNA) 261, a VGA262, an LO mixer 264A for the I signal path, amplifier/filter circuitry266A for the I signal path, an analog-to-digital converter (ADC) 268Afor the I signal path, an LO mixer 264B for the Q signal path,amplifier/filter circuitry 266B for the Q signal path, and an ADC 268Bfor the Q signal path. The mixers 264A and 264B directly down-convertthe received signal into baseband I and Q signals by mixing the receivedsignal with local oscillator signals LO(I) and LO(Q), where thefrequency of the local oscillator signals (as generated by a localoscillator, not shown) is ideally the carrier frequency. Mismatchbetween mixers 264A and 264B, between amplifiers/filters 266A and 266B,and/or between ADCs 268A and 268B results in receiver-side I/Q mismatch.A difference between the frequency of the local oscillator signals inthe receiver unit 250 of a receiver 106 (FIG. 1A) and the correspondingfrequency of local oscillator signals in the transmitter unit 210 of atransmitter 102 (FIG. 1A) results in carrier frequency offset.

The components described with reference to FIG. 2A are exemplary only.In various embodiments, one or more of the components described may beomitted, combined, or modified, and additional components may beincluded. For instance, in some embodiments, the transmitter unit 210and receiver unit 250 may share a common antenna, or may have variousadditional antennas and transmitter/receiver chains. In otherembodiments, there may be no antenna; instead, the transmitter unit 210and receiver unit 250 connect to a wired link. In some implementations,the transceiver 200 may include less or more filter and/or amplifiercircuitry (e.g., blocks 222 and 266 of FIG. 2A).

Attention is now directed to mathematical modeling of I/Q mismatch. FIG.2B illustrates I/Q mismatch in a transmitter front end (e.g.,transmitter AFE 220, FIG. 2A) in accordance with some embodiments. Asignal component x_(R)[n] is provided to the I signal path, whichincludes components 270A (e.g., DAC 221A and amplifier/filter 222A, FIG.2A) and a mixer 224A. A signal x_(I)[n] is provided to the Q signalpath, which includes components 270B (e.g., DAC 221B andamplifier/filter 222B, FIG. 2A) and a mixer 224B. (The R and Isubscripts refer to real and imaginary components, and thus respectivelyto the I and Q components, of a signal x[n].) The signal path components270A and 270B have corresponding functions f[n]+g[n] and f[n]−g[n],respectively, where f[n] is common between the components 270A and 270Band g[n] includes an amplitude mismatch (e.g., a frequency-dependentamplitude mismatch) between components 270A and 270B. A phase mismatchΔφ is introduced during up-conversion by mixers 224A and 224B. The I/Qmismatch of FIG. 2B thus is represented by g[n] and Δφ. The combiner 272combines the up-converted I and Q signal to generate an RF transmittedsignal.

The baseband equivalent of the RF transmitted signal, as affected by theI/Q mismatch of FIG. 2B, is

r[n]=[cos Δφ(f[n]+g[n])*x _(R) [n]−sin Δφ(f[n]−g[n])*x _(I) [n]]+j[−sinΔφ( f[n]+g[n])*x _(R) [n]+cos Δφ(f[n]−g[n])*x _(I) [n]]

With some change in notation, this expression may be written in matrixform as

$\begin{bmatrix}{r_{R}\lbrack n\rbrack} \\{r_{I}\lbrack n\rbrack}\end{bmatrix} = {\begin{bmatrix}{\cos \; \Delta \; {\varphi \left( {{f\lbrack n\rbrack} + {g\lbrack n\rbrack}} \right)}} & {{- \sin}\; \Delta \; {\varphi \left( {{f\lbrack n\rbrack} - {g\lbrack n\rbrack}} \right)}} \\{{- \sin}\; \Delta \; {\varphi \left( {{f\lbrack n\rbrack} + {g\lbrack n\rbrack}} \right)}} & {\cos \; \Delta \; {\varphi \left( {{f\lbrack n\rbrack} - {g\lbrack n\rbrack}} \right)}}\end{bmatrix}*\begin{bmatrix}{x_{R}\lbrack n\rbrack} \\{x_{I}\lbrack n\rbrack}\end{bmatrix}}$

Expanding the matrix results in

$\begin{bmatrix}{r_{R}\lbrack n\rbrack} \\{r_{I}\lbrack n\rbrack}\end{bmatrix} = {{\begin{bmatrix}{{f\lbrack n\rbrack}\cos \; \Delta \; \varphi} & {{g\lbrack n\rbrack}\; \sin \; \Delta \; \varphi} \\{{- {g\lbrack n\rbrack}}\sin \; \Delta \; \varphi} & {{f\lbrack n\rbrack}\cos \; \Delta \; \varphi}\end{bmatrix}*\begin{bmatrix}{x_{R}\lbrack n\rbrack} \\{x_{I}\lbrack n\rbrack}\end{bmatrix}} + {\quad{\begin{bmatrix}{{g\lbrack n\rbrack}\cos \; \Delta \; \varphi} & {{f\lbrack n\rbrack}\sin \; \Delta \; \varphi} \\{{- {f\lbrack n\rbrack}}\sin \; \Delta \; \varphi} & {{g\lbrack n\rbrack}\cos \; \Delta \; \varphi}\end{bmatrix}*\begin{bmatrix}{x_{R}\lbrack n\rbrack} \\{- {x_{I}\lbrack n\rbrack}}\end{bmatrix}}}}$

This equation corresponds to the following relation in complex notation:

r[n]=(f[n] cos Δφ−jg[n] sin Δφ)*x[n]+(g[n] cos Δφ−jf[n] sin Δφ)*x*[n]

If we define

a[n]=f[n] cos Δφ−jg[n] sin Δφ

b[n]=g[n] cos Δφ−jf[n] sin Δφ

then we can write compactly

r[n]=a[n]*x[n]+b[n]*x*[n]

As this result indicates, I/Q mismatch causes interference between I andQ components in the time domain. Equivalently, I/Q mismatch causesinterference between mirror frequencies in the frequency domain, asshown by transforming the equation for r[n] into the frequency domain:

R(f)=A(f)X(f)+B(f)X*(−f).

I/Q mismatch can be perfectly compensated in the frequency domain viathe following linear combination of the signal R(f) and its conjugate:

$\begin{matrix}{{Y(f)} = {{A*\left( {- f} \right){R(f)}} - {{B(f)}R*\left( {- f} \right)}}} \\{= {\left\lbrack {{A*\left( {- f} \right){A(f)}} - {B*\left( {- f} \right){B(f)}}} \right\rbrack {X(f)}}}\end{matrix}$

where {tilde over (H)}(f)=A*(−f)A(f)−B*(−f)B(f) is the equivalenttransmitter shaping filter, which acts as a scaling factor to becorrected for. Dividing Y(f) by {tilde over (H)}(f) recovers theoriginal signal X(f).

This expression for the compensated signal Y(f) can be simplified byexpressing complex signals in terms of their real and imaginary parts.Also, it is possible to define an alternative correction formula,

$\begin{matrix}{{Y(f)} = {{R(f)} - {\frac{B(f)}{A^{*}\left( {- f} \right)}R*\left( {- f} \right)}}} \\{= {{A(f)}\left( {1 - \frac{{B(f)}{B^{*}\left( {- f} \right)}}{{A(f)}{A^{*}\left( {- f} \right)}}} \right){X(f)}}}\end{matrix}$

Note that the equivalent shaping filter is different in this case.

If the I/Q mismatch is small, we can approximate f[n]≈1, sin Δφ≈Δφ, cosΔφ≈1, g[n] sin Δφ≈0, so that the complex envelope of the transmittedsignal reads

r[n]=x[n]+b[n]*x*[n],

where b[n]=g[n]−jΔφ. Also, the interference due to I/Q mismatch can besubtracted directly as y[n]=r[n] b[n]*x*[n].

The above mathematics model I/Q mismatch in the transmitter (e.g.,transmitter 102, FIG. 1A). Receiver-side I/Q mismatch (e.g., in thereceiver 106, FIG. 1A) may be modeled in the same fashion.

Assuming the only sources of distortion in the communication system aretransmitter (Tx) I/Q mismatch, receiver (Rx) I/Q mismatch, andmulti-path effects in the channel, the received signal in the frequencydomain is

Z(f)=Ã(f)X(f)+{tilde over (B)}(f)X*(−f)+A _(Rx) W(f)+B _(Rx)(f)W*(−f)

where

Ã(f)=A _(Rx)(f)H(f)A _(Tx)(f)+B _(Rx)(f)H*(−f)B _(Tx)*(−f)

{tilde over (B)}(f)=A _(Rx)(f)H(f)B _(Tx)(f)+B _(Rx)(f)H*(−f)A_(Tx)*(−f).

W(f) is the spectrum of the additive Gaussian noise and H(f) is thefrequency response of the channel. (H(f) is thus unrelated to {tildeover (H)}(f)). The corrected signal {circumflex over (Z)}(f) is definedas

${\hat{Z}(f)} = {{X(f)} - {\frac{\overset{\sim}{B}(f)}{{\overset{\sim}{A}}^{*}\left( {- f} \right)}{X^{*}\left( {- f} \right)}}}$

After straightforward calculations, it can be shown that the correctedsignal reads

${\hat{Z}(f)} = {{{\overset{\sim}{A}(f)}\left( {1 - \frac{{\overset{\sim}{B}(f)}{{\overset{\sim}{B}}^{*}\left( {- f} \right)}}{{\overset{\sim}{A}(f)}{{\overset{\sim}{A}}^{*}\left( {- f} \right)}}} \right){X(f)}} - {\overset{\sim}{W}(f)}}$

where {tilde over (W)}(f) is the equivalent noise after the correctionfilter has been applied.

In some embodiments, transmitted signals are narrowband signals. Fornarrowband signals, the filters in the I and Q signal paths (e.g.,filters 222A-B and 266A-B, FIG. 2A) function as scalar multipliers. Forexample, functions a[n] and b[n] reduce to scalars a and b. Also, signalimpairments due to multi-path effects in the channel may be neglected innarrowband embodiments.

FIG. 2C illustrates signal impairments in a system (e.g., system 100,FIG. 1A) in which the received signal is affected by AWGN, carrierfrequency offset 126, and receiver-side I/Q mismatch 128. Noise w[n] ismixed into the signal x[n] in the channel.

The received sampled signal z[n] can be expressed as

z[n]=a[e ^(jΔωn) x[n]]+b[e ^(−jΔωn) x*[n]]+aw[n]+bw*[n]

where Δω is the carrier frequency offset (CFO) 126. The Rx I/Q mismatch128 creates two different components rotating in opposite direction.

There are algorithms for CFO estimation based on the autocorrelation ofthe received signal, r_(zz)[M]=z[n+M]z*[n]. The autocorrelation at lag Mfor the received signal affected by Rx I/Q mismatch 128 is

r _(zz) [M]=|a| ² |x[n]| ² e ^(jΔωM) +|b| ² |x[n]| ² e ^(−jΔωM) +ab*e^(j2Δω(n+M))(x[n])² +a*be ^(−j2Δω(n+M))(x*[n])²

The second term significantly degrades the accuracy of CFO estimationbased on autocorrelation of the received signal.

However, the CFO 126 may be estimated using other estimation techniques.For example, the CFO 126 may be estimated using classical non-linearleast squares (NLS) techniques in receivers that process multiplesymbols at a time (e.g., when the system employs a repeating trainingsequence or a cyclic prefix of an OFDM symbol, which repeats twice andchanges from OFDM symbol to OFDM symbol).

Once Δω is known (or estimated), the (scalar) I/Q mismatch parameters a,b may be estimated. Defining {tilde over (z)}[n]=e^(jΔωn)x[n], we canwrite the following matrix equation:

$\begin{bmatrix}{z\lbrack n\rbrack} \\{z\left\lbrack {n + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\sim}{x}\lbrack n\rbrack} & {\overset{\sim}{x}*\lbrack n\rbrack} \\{\overset{\sim}{x}\left\lbrack {n + 1} \right\rbrack} & {\overset{\sim}{x}*\left\lbrack {n + 1} \right\rbrack}\end{bmatrix}\begin{bmatrix}a \\b\end{bmatrix}} + \begin{bmatrix}{\overset{\sim}{w}\lbrack n\rbrack} \\{\overset{\sim}{w}\left\lbrack {n + 1} \right\rbrack}\end{bmatrix}}$

The I/Q imbalance parameters can then be estimated by inverting the 2×2system matrix in this equation.

The accuracy of the estimate of the I/Q mismatch parameters can beimproved by stacking samples received over a whole period of M samples.Defining the input signal vector {tilde over (x)}[n]=[{tilde over(x)}[n] {tilde over (x)}[n+1] . . . {tilde over (x)}[n+M−1]] and thereceived signal vector z[n]=[z[n] z[n+1] . . . z[n+M−1]], the receivedsignal vector may be rewritten as

${z\lbrack n\rbrack} = {{\begin{bmatrix}{\overset{\sim}{x}\lbrack n\rbrack} & {{\overset{\sim}{x}}^{*}\lbrack n\rbrack}\end{bmatrix}\begin{bmatrix}a \\b\end{bmatrix}} + {\overset{\sim}{w}\lbrack n\rbrack}}$

Solving for a, b now entails the inversion of an M×2 matrix. However,because of the linearity of the problem, classical low-complexityadaptive filtering techniques, such as least mean squares (LMS), may beused.

Once the I/Q mismatch parameters a, b have been found, the Rx I/Qmismatch 128 can then be corrected by a linear combination of z[n] andz*[n],

$\begin{matrix}{{\hat{z}\lbrack n\rbrack} = {{a^{*}{z\lbrack n\rbrack}} - {{bz}^{*}\lbrack n\rbrack}}} \\{= {{{^{j\; \Delta \; \omega \; n}\left( {{a}^{2} - {b}^{2}} \right)}{x\lbrack n\rbrack}} + {w\lbrack n\rbrack}}}\end{matrix}$

As this equation indicates, the CFO 126 can now be corrected using phaserotation.

In some embodiments, iterative estimations of the CFO 126 and Rx I/Qmismatch 128 are performed. FIG. 3A illustrates a method 350 ofiteratively estimating CFO and I/Q mismatch. The method 350 is performedin a receiver (e.g., receiver unit 250 of transceiver 200, FIG. 2A) inaccordance with some embodiments.

In the method 350, a repeating or periodic signal is received (351). Insome embodiments, the signal is a repeating training sequence. In someembodiments, the signal is a narrowband signal. In some embodiments, thesignal is a cyclic prefix of an OFDM symbol.

CFO (e.g., CFO 126, FIG. 2C) is estimated (352) based on the signalassuming no I/Q mismatch. For example, the CFO is estimated using anautocorrelation technique; the iterative nature of the method 350accommodates the inaccuracy associated with using autocorrelation.Alternatively, the CFO is estimated using an NLS technique based on thesignal. Using the estimated CFO, I/Q mismatch (e.g., receiver-side I/Qmismatch 128, FIG. 2C) is estimated (354). The estimated I/Q mismatch iscompensated for (356), and the CFO is then re-estimated (358). The I/Qmismatch is re-estimated (360) using the re-estimated CFO and iscompensated for (362), and the re-estimated CFO is compensated for(364).

While the method 350 includes a number of operations that appear tooccur in a specific order, it should be apparent that the method 350 caninclude more or fewer operations. An order of two or more operations maybe changed and two or more operations may be combined into a singleoperation.

In some embodiments, the method 350 is an example of a first phase of atwo-phase process of estimating and compensating for signal impairments.In the first phase, estimation of and compensation for receiver-side I/Qmismatch and carrier frequency offset are performed. In a subsequentsecond phase, estimation of and compensation for transmitter-side I/Qmismatch and channel distortion (e.g., linear channel distortion) areperformed. Impairments 126 and 128 (FIG. 1B) thus are estimated andcompensated for in the first phase, and impairments 122 and 124 (FIG.1B) are estimated and compensated for in the second phase. Thistwo-phase approach provides for computational simplicity compared toapproaches that jointly estimate and correct for transmitter-side I/Qmismatch, receiver-side I/Q mismatch, channel distortion, and carrierfrequency offset. The two-phase approach thus is easier to implementthan joint approaches.

In some embodiments, a two-phased approach is used in communicationssystems in which the physical layer employs a repetitive known signal(e.g., a training sequence, preamble, or prefix). For example, thetwo-phased approach may be implemented in systems compatible with one ofthe IEEE 802.11 family of protocols. In some embodiments, a two-phasedapproach is used in communications systems that perform multicarriermodulation based on orthogonal frequency-division multiplexing (OFDM).Examples include systems compatible with one of the IEEE 802.11 familyof protocols and systems compatible with the 3GPP E-UTRAN (LTE)standard.

FIG. 3B is a flowchart illustrating a two-phased method 300 ofestimating and compensating for signal impairments in accordance withsome embodiments. The method 300 is performed in a receiver (e.g.,receiver 106, FIG. 1A). In some embodiments, the method 300 is performedby the receiver baseband processor 280 (FIG. 2A) (e.g., by the signalimpairment compensation unit 285, FIG. 2A). In some embodiments, themethod 300 uses a training signal or dedicated repeating preamble but ingeneral the method 300 is not so limited. The method 300 thus isperformed in the digital domain in baseband in accordance with someembodiments. During phase one, carrier frequency offset (e.g., CFO 126,FIG. 2C) and receiver-side I/Q mismatch (e.g., Rx I/Q mismatch 128, FIG.2C) are repeatedly estimated for a predefined number of iterations.Phase one is terminated, however, if the estimated carrier frequencyoffset is determined to be less than a specified threshold, even if thepredefined number of iterations has not been completed. In response to adetermination that the estimated carrier frequency offset is less thanthe specified threshold determination, the method 300 proceeds to phasetwo, in which channel distortion is estimated and equalized andtransmitter-side I/Q mismatch is estimated and compensated for.

At the start 302 of the method 300, an iteration counter (N_iter) is setto zero. An estimate of carrier frequency offset is made (304) using anyknown technique. For example, non-linear least-squares (NLS) techniquesthat process multiple symbols at a time (e.g., from a repeating trainingsequence or the cyclic prefix of an OFDM symbol) may be used to estimatethe carrier frequency offset. Alternately, autocorrelation techniquesmay be used. In some embodiments, the estimation of carrier frequencyoffset is made assuming no receiver-side I/Q mismatch. For example, theestimation of carrier frequency offset is agnostic toward receiver-sideI/Q mismatch.

The estimated carrier frequency offset is compared (306) to a predefinedthreshold (CFO_thresh). The predefined threshold is system-dependent. Insome embodiments implemented in OFDM systems, the predefined thresholdis set to be comparable to the sub-carrier spacing.

If the estimated carrier frequency offset is less than the predefinedthreshold (306—Yes), phase one is terminated and the method 300 proceedsto operations 320 and 322 of phase two (described below).

If the estimated carrier frequency offset is not less than thepredefined threshold (306—No), an estimate is made (308) of thereceiver-side I/Q mismatch. This estimate is made, for example, asdescribed below with regard to FIGS. 4A-4D and equations (1)-(5).

The iteration counter is incremented (310) and compared (312) to thepredefined number of iterations (N_max). In some embodiments, N_maxequals two. In some embodiments, N_max has a value in the range of 2-10.

If the incremented iteration counter is not greater than N_max (312—No),compensation is performed (314) for the receiver-side I/Q mismatchestimated at 308, and the method 300 returns to operation 304.

If the incremented iteration counter is greater than N_max (312—Yes),compensation is performed (316) for the receiver-side I/Q mismatchestimated at 308, and compensation is performed (318) for carrierfrequency offset estimated at 304. The carrier frequency offsetcompensation is performed using any known technique (e.g., phaserotation). At this point, phase one is complete. The estimationoperations 304 and 308 and compensation operations 314/316 thus areperformed a number of times equal to N_max, assuming that the estimatedcarrier frequency offset is not determined to be less than thepredefined threshold during one of the iterations. Note that if N_max=2and the determination at 306 is “No” in both iterations, phase one is anexample of the method 350 (FIG. 3A). If N_max (i.e., the predefinednumber of iterations) is greater than two, portions of phase one are anexample of the method 350 (FIG. 3A): for example, operations 352-356 ofthe method 350 may correspond to a first iteration and operations358-362 of the method 350 may correspond to a final iteration.

In phase two of method 300, a joint estimate is made (320) of channeldistortion (e.g., distortion 124, FIG. 1B) and transmitter-side I/Qmismatch (e.g., mismatch 122, FIG. 1B). Joint equalization of theestimated channel distortion and compensation for the estimatedtransmitter-side I/Q mismatch is performed (322), at which point themethod 300 ends (324). Equalization of channel distortion andcompensation for transmitter-side I/Q mismatch thus are performed afterreceiver-side I/Q mismatch and carrier frequency offset have beencompensated for in accordance with some embodiments. (Equalization ofchannel distortion compensates for the channel distortion.) In someembodiments, equalization of channel distortion and compensation fortransmitter-side I/Q mismatch are performed in the frequency domain, asdescribed below.

While the method 300 includes a number of operations that appear tooccur in a specific order, it should be apparent that the method 300 caninclude more or fewer operations. An order of two or more operations maybe changed and two or more operations may be combined into a singleoperation.

Attention is now directed to estimating and compensating forreceiver-side I/Q mismatch. A repeating or periodic narrowband signal(e.g., a training signal) z[n] is used, as shown in FIG. 2D and FIG. 4Ain accordance with some embodiments, and the carrier frequency offset isestimated beforehand (e.g., the CFO is estimated in operation 304, FIG.3B, prior to estimating the receiver-side I/Q mismatch in operation308). The variable n indexes samples of the signal. Each period 402 ofthe signal z[n] includes M samples, as shown in FIG. 4A, where M is aninteger greater than or equal to one. For example, a first period 402-1includes M samples, as does a second period 402-2, a third period 402-3,and an Nth period 402-N.

The signal z[n] may be expressed as:

z[n]=(a _(Rx) a _(Tx))e ^(jΔωn) x[n]+(a _(Rx) b _(Tx))e ^(jΔωn) x*[n]+(b_(Rx) a _(Tx)*)e ^(−jΔωn) x*[n]+( b _(Rx) b _(Tx)*)e ^(−jΔωn) x[n]+a_(Rx) w[n]+b _(Rx) w*[n]

The received signal over two successive periods of length M is:

z[n]=a _(Rx) [e ^(jΔωn) y[n]]+b _(Rx) [e ^(−jΔωn) y*[n]]+{tilde over(w)}[n]

z[n+M]=a _(Rx) e ^(jΔωM) [e ^(jΔωn) y[n]]+b _(Rx) e ^(−jΔωM) [e ^(−jΔωn)y*[n]]+{tilde over (w)}[n+M]

The signal y[n] includes the (unknown) transmitter-side I/Q mismatch 122(FIG. 2D).

Receiver-side I/Q mismatch can be compensated by performing a lineartransformation involving a scalar correction factor q to generate acompensated received signal {circumflex over (z)}[n]. Specifically:

$\begin{matrix}{{\hat{z}\lbrack n\rbrack} = {\begin{bmatrix}{z\lbrack n\rbrack} & {z^{*}\lbrack n\rbrack}\end{bmatrix}\begin{bmatrix}1 \\q\end{bmatrix}}} & (1)\end{matrix}$

where z*[n] is the complex conjugate of z[n].

Assuming perfect compensation and thus perfect correction,

$q = \frac{- b_{RX}}{a_{Rx}^{*}}$

and the corrected received signal over two successive training sequencesreads

$\mspace{20mu} {{z\lbrack n\rbrack} = {{{a_{Rx}\left( {1 - \frac{{b_{Rx}}^{2}}{{a_{Rx}}^{2}}} \right)}^{j\; \Delta \; \omega \; n}{y\lbrack n\rbrack}} + {{a_{Rx}\left( {1 - \frac{{b_{Rx}}^{2}}{{a_{Rx}}^{2}}} \right)}{w\lbrack n\rbrack}}}}$${z\left\lbrack {n + M} \right\rbrack} = {{a_{Rx}{^{j\; \Delta \; \omega \; M}\left( {1 - \frac{{b_{Rx}}^{2}}{{a_{Rx}}^{2}}} \right)}^{j\; \Delta \; \omega \; n}{y\lbrack n\rbrack}} + {{a_{Rx}\left( {1 - \frac{{b_{Rx}}^{2}}{{a_{Rx}}^{2}}} \right)}{w\left\lbrack {n + M} \right\rbrack}}}$

Neglecting the noise and assuming perfect compensation for receiver-sideI/Q mismatch, it holds that:

{circumflex over (z)}[n+M]=e ^(jΔωM) {circumflex over (z)}[n]  (2)

where Δω is the carrier frequency offset (e.g., as estimated inoperation 304, FIG. 3B) in units of radians/sample. Relationship (2)thus holds that a compensated sample in a given period (e.g., period402-2) equals the compensated sample in a previous period (e.g., period402-1) multiplied by a phase factor determined by the product of thecarrier frequency offset and the number of samples in a period.

Relationship (2) holds for all samples within two successive periods.FIG. 4B illustrates samples in two successive periods, including avector z₁ of samples in period 402-1 and a vector z₂ of samples inperiod 402-2. Relationship (2) holds between vectors z₁ and z₂, suchthat:

$\begin{matrix}{{\left\lceil \begin{matrix}z_{2} & z_{2}^{*}\end{matrix} \right\rceil \begin{bmatrix}1 \\q\end{bmatrix}} = {{^{{j\Delta\omega}\; M}\begin{bmatrix}z_{1} & z_{1}^{*}\end{bmatrix}}\begin{bmatrix}1 \\q\end{bmatrix}}} & (3)\end{matrix}$

Solving equation (3) for the correction factor q yields:

$\begin{matrix}{q = \frac{{- \left( {z_{2}^{*} - {^{{j\Delta\omega}\; M}z_{1}^{*}}} \right)^{H}}\left( {z_{2} - {^{{j\Delta\omega}\; M}z_{1}}} \right)}{{{z_{2}^{*} - {^{{j\Delta\omega}\; M}z_{1}^{*}}}}^{2}}} & (4)\end{matrix}$

In equation (4), H refers to the Hermitian transpose operation (i.e.,taking the transpose conjugate of the vector by transposing the vectorand taking the complex conjugate of each element). The correction factorq is determined using equation (4) and is then used to determine thecompensated received signal {circumflex over (z)}[n] in accordance withequation (1).

In some embodiments, accuracy of the receiver-side I/Q mismatchestimation is improved by averaging multiple estimates from adjacentperiods. For example, a first estimate is made based on periods 402-1and 402-2 and a second estimate is made based on periods 402-3 and402-4, as illustrated in FIG. 4C in accordance with some embodiments.Each of the two estimates is made, for example, using equation (4). Thefirst and second estimates are then averaged, and the resulting value ofq is used to compensate for the receiver-side I/Q mismatch usingequation (1).

In some embodiments, conditioning of the problem of estimating thereceiver-side I/Q mismatch is improved by using non-consecutive periods.Using non-consecutive periods helps to ensure a reasonably large valueof the carrier frequency offset (Δω). FIG. 4D illustrates an example inwhich vectors z₁ and z_(k) for respective non-consecutive periods 402-1and 402-k are used, where the index k indicates the separation betweenthe two periods (i.e., there are k−1 periods between period 402-1 andperiod 402-k). In this example, equation (3) is modified to:

$\begin{matrix}{{\left\lceil \begin{matrix}z_{k} & z_{k}^{*}\end{matrix} \right\rceil \begin{bmatrix}1 \\q\end{bmatrix}} = {{^{{j\Delta\omega}\; k\; M}\begin{bmatrix}z_{1} & z_{1}^{*}\end{bmatrix}}\begin{bmatrix}1 \\q\end{bmatrix}}} & (5)\end{matrix}$

Equation (5) is then solved for the correction factor q, and theresulting value of q is used to compensate for the receiver-side I/Qmismatch in accordance with equation (1).

In some embodiments, instead of using a periodic training signal, thecyclic prefix of an OFDM symbol is used to estimate the receiver-sideI/Q mismatch. While the cyclic prefix changes from OFDM symbol to OFDMsymbol, it repeats twice and thus may be considered a repeating signal.

Equations (1)-(5) describe performing estimation of and compensation forreceiver-side I/Q mismatch in the time domain. In some embodiments,after time-domain compensation of receiver-side I/Q mismatch (e.g., inphase one of method 300, FIG. 3B), transmitter-side I/Q mismatch andchannel distortion are compensated for in the frequency domain (e.g., inphase two of method 300, FIG. 3B). The frequency-domain compensation mayalso compensate for residual receiver-side I/Q mismatch that was notcompensated for in the time domain. Frequency-domain compensation isperformed, for example, in the receiver of an OFDM system.

In the frequency domain, OFDM transmissions may be modeled by a complextransmit symbol vector x_(c), a complex diagonal channel matrix H_(c), acomplex receive symbol vector y_(c), and a complex additive noise vectorn_(c):

y _(c) =H _(c) x _(c) +n _(c).

An equivalent real-valued notation is:

y = Hx + n where $x = {{x_{cR} \otimes \begin{bmatrix}1 \\0\end{bmatrix}} + {x_{cI} \otimes \begin{bmatrix}0 \\1\end{bmatrix}}}$ $n = {{n_{cR} \otimes \begin{bmatrix}1 \\0\end{bmatrix}} + {n_{cI} \otimes \begin{bmatrix}0 \\1\end{bmatrix}}}$ $y = {{y_{cR} \otimes \begin{bmatrix}1 \\0\end{bmatrix}} + {y_{cI} \otimes \begin{bmatrix}0 \\1\end{bmatrix}}}$ and $H = {{H_{cR} \otimes \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}} + {H_{cI} \otimes {\begin{bmatrix}0 & {- 1} \\1 & 0\end{bmatrix}.}}}$

Here,

denotes the Kronecker product. H becomes a block diagonal matrix withblocks of size 2×2, as illustrated in FIG. 6 in accordance with someembodiments. The real and imaginary parts of each entry are stacked ontop of each other in x, y, and n.

Transmitter IQ mismatch and receiver IQ mismatch cause interferencebetween a frequency f and its mirror frequency −f. The real-valued modelallows us to redefine H such that it includes the modeling oftransmitter IQ mismatch and receiver IQ mismatch. FIG. 6 illustrates thestructure of the effective channel matrix H including transmitter IQmismatch and receiver IQ mismatch. The interference between f and f addsblocks on the main skew diagonal of H. In the center, the 2×2 matrix 600corresponds to the zero frequency. There is no interference betweensymbols that are affected by entries with different fill patterns in thematrix. For example, symbols affected by entries 602-1 through 602-4 donot interfere with symbols affected by entries 600, 604-1 through 604-4,and 606-1 through 606-4, and so on. Therefore, it is sufficient toconsider a submatrix composed of the entries marked with a single fillpattern in FIG. 6.

The transmitter IQ offsets and receiver IQ offsets are respectivelymodeled by

$\begin{bmatrix}{\cos \left( \phi_{T} \right)} & {A_{T}{\sin \left( \phi_{T} \right)}} & {{- A_{T}}{\cos \left( \phi_{T} \right)}} & {\sin \left( \phi_{T} \right)} \\{{- A_{T}}{\sin \left( \phi_{T} \right)}} & {\cos \left( \phi_{T} \right)} & {\sin \left( \phi_{T} \right)} & {A_{T}{\cos \left( \phi_{T} \right)}} \\{{- A_{T}}{\cos \left( \phi_{T} \right)}} & {\sin \left( \phi_{T} \right)} & {\cos \left( \phi_{T} \right)} & {A_{T}{\sin \left( \phi_{T} \right)}} \\{\sin \left( \phi_{T} \right)} & {A_{T}{\cos \left( \phi_{T} \right)}} & {{- A_{T}}{\sin \left( \phi_{T} \right)}} & {\cos \left( \phi_{T} \right)}\end{bmatrix}$ ${and}\begin{bmatrix}{\cos \left( \phi_{R} \right)} & {{- A_{R}}{\sin \left( \phi_{R} \right)}} & {A_{R}{\cos \left( \phi_{R} \right)}} & {\sin \left( \phi_{R} \right)} \\{A_{R}{\sin \left( \phi_{R} \right)}} & {\cos \left( \phi_{R} \right)} & {\sin \left( \phi_{R} \right)} & {{- A_{R}}{\cos \left( \phi_{R} \right)}} \\{{- A_{R}}{\cos \left( \phi_{R} \right)}} & {\sin \left( \phi_{R} \right)} & {\cos \left( \phi_{R} \right)} & {A_{R}{\sin \left( \phi_{R} \right)}} \\{\sin \left( \phi_{R} \right)} & {A_{R}{\cos \left( \phi_{R} \right)}} & {{- A_{R}}{\sin \left( \phi_{R} \right)}} & {\cos \left( \phi_{R} \right)}\end{bmatrix}$

where A_(T)/2 and A_(R)/2 are respectively the transmitter and receiveramplitude offsets and φ_(T)/2 and φ_(R)/2 are respectively thetransmitter and receiver phase offsets in terms of percent. Assumingthat the frequency response is represented by the complex numbers h_(f)and h_(−f) we define a symmetric component

h _(s)=(h _(f) +h _(−f))/2=|h _(s)|(cos(φ_(cs))+j sin(φ_(cs)))

and an asymmetric component

h _(a)=(h _(f) −h _(−f))/2=|h _(a)|(cos(φ_(ca))+j sin(φ_(ca)))

Combining the transmitter IQ offset, the channel matrix, and thereceiver IQ offset, we obtain

$\begin{matrix}{Z = \begin{bmatrix}Z_{1} & {- Z_{2}} & Z_{7} & Z_{8} \\Z_{2} & Z_{1} & Z_{8} & {- Z_{7}} \\Z_{3} & Z_{4} & Z_{5} & {- Z_{6}} \\Z_{4} & {- Z_{3}} & Z_{6} & Z_{5}\end{bmatrix}} \\{= {\begin{bmatrix}X_{1} & {- X_{2}} & X_{3} & X_{4} \\X_{2} & X_{1} & X_{4} & {- X_{3}} \\X_{3} & X_{4} & X_{1} & {- X_{2}} \\X_{4} & {- X_{3}} & X_{2} & X_{1}\end{bmatrix} + \begin{bmatrix}Y_{1} & {- Y_{2}} & {- Y_{3}} & {- Y_{4}} \\Y_{2} & Y_{1} & {- Y_{4}} & Y_{3} \\Y_{3} & Y_{4} & {- Y_{1}} & Y_{2} \\Y_{4} & {- Y_{3}} & {- Y_{2}} & {- Y_{1}}\end{bmatrix}}}\end{matrix}$with

X ₁ =|h _(s)|[cos(φ_(cs))(1+A _(R) A _(T))cos(φ_(T)−φ_(R))−sin(φ_(cs))(A_(T) +A _(R))sin(φ_(T)−φ_(R))],

Y ₁ =|h _(a)|[cos(φ_(ca))(1−A _(R) A _(T))cos(φ_(T)+φ_(R))−sin(φ_(ca))(A_(T) −A _(R))sin(φ_(T)+φ_(R))],

X ₂ =|h _(s)|[cos(φ_(cs))(A _(T) −A_(R))sin(φ_(T)+φ_(R))+sin(φ_(cs))(1−A _(R) A _(T))cos(φ_(T)+φ_(R))],

Y ₂ =|h _(a)|[cos(φ_(ca))(A _(T) +A_(R))sin(φ_(T)−φ_(R))+sin(φ_(ca))(1+A _(R) A _(T))cos(φ_(T)−φ_(R))],

X ₃ =|h _(s)|[cos(φ_(cs))(A _(T) +A_(R))cos(φ_(T)−φ_(R))−sin(φ_(cs))(1+A _(R) A _(T))sin(φ_(T)+φ_(R))],

Y ₃ =|h _(a)|[cos(φ_(ca))(A _(T) −A_(R))cos(φ_(T)+φ_(R))−sin(φ_(ca))(1−A _(R) A _(T))sin(φ_(T)+φ_(R))],

X ₄ =|h _(s)|[cos(φ_(cs))(1−A _(R) A _(T))sin(φ_(T)+φ_(R))+sin(φ_(cs))(A _(T) −A _(R))cos(φ_(T)+φ_(R))],

and

Y ₄ =|h _(a)|[cos(φ_(ca))+(1+A _(R) A_(T))sin(φ_(T)−φ_(R))+sin(φ_(ca))(A _(T) +A _(R))cos(φ_(T)−φ_(R))].

Note that X denotes the contribution due to h_(s) and Y denotes thecontribution due to h_(a).

An estimate of H may be obtained by estimating the frequency response atcertain frequencies and subsequently interpolating between theseestimates. The estimated frequency responses are spaced closely enoughwith respect to the coherence bandwidth. The frequency response isjointly estimated for f and its mirror frequency −f. This is done usingpilot symbols that are symmetric with respect to frequency zero.Considering Z we observe that eight parameters are to be estimatedjointly. At least two OFDM symbols are used to get a valid estimate.Examples of valid pilot matrices include

${P = \begin{bmatrix}1 & {- 1} \\0 & 0 \\1 & 1 \\0 & 0\end{bmatrix}},{P = \begin{bmatrix}1 & 0 \\0 & 0 \\0 & 1 \\0 & 0\end{bmatrix}},$

$\begin{matrix}{P = \begin{bmatrix}p_{1} & {- p_{3}} \\p_{2} & {- p_{4}} \\p_{3} & p_{1} \\p_{4} & p_{2}\end{bmatrix}} & (6)\end{matrix}$

or more generallywhere the first column specifies a symbol transmitted in time slot 1 andthe second column specifies a symbol transmitted in time slot 2. Also,every entry in a respective column may be multiplied by −1 withoutchanging the relevant properties. In some embodiments, the interpolationis done independently for the diagonal elements and the skew diagonalelements.

Assuming that P was transmitted, R=ZP is observed at the receiver. Basedon P and

$R = \begin{bmatrix}r_{1} & r_{5} \\r_{2} & r_{6} \\r_{3} & r_{7} \\r_{4} & r_{8}\end{bmatrix}$

we construct the following matrices

$R_{eff} = \begin{bmatrix}r_{1} & {- r_{2}} & r_{5} & {- r_{6}} \\r_{2} & r_{1} & r_{6} & r_{5} \\r_{3} & r_{4} & r_{7} & r_{8} \\r_{4} & {- r_{3}} & r_{8} & {- r_{7}}\end{bmatrix}$

and the orthogonal pilot matrix

$P_{eff} = {\begin{bmatrix}p_{1} & {- p_{2}} & {- p_{3}} & p_{4} \\p_{2} & p_{1} & {- p_{4}} & {- p_{3}} \\p_{3} & p_{4} & p_{1} & p_{2} \\p_{4} & {- p_{3}} & p_{2} & {- p_{1}}\end{bmatrix}.}$

From this we can estimate Z:

Z _(est) =R _(eff) P _(eff) ^(H)(P _(eff) *P _(eff) ^(H))⁻¹.  (7)

Note that (P_(eff)*P_(eff) ^(H))⁻¹ is a diagonal matrix and thus can berepresented as a scaling factor. To apply a zero-forcing approach,Z_(est) is inverted. The inversion of Z has the same complexity as theinversion of a 2×2 complex matrix even though there does not exist acomplex-valued equivalent. Hence, no inversions of arbitrary 4×4matrices are involved in compensating for Z.

This frequency-domain compensation technique includes the estimation andcorrection of a frequency-dependent IC) offset. In particular,frequency-selective IC) offset compensation is performed at the edgefrequencies.

Also, this frequency-domain compensation technique assumes no carrierfrequency offset (e.g., as for the phase two of method 300, FIG. 3B). Ifthere were a carrier frequency offset, H would result in a fullyoccupied matrix, which reflects the inter-subcarrier interference causedby the carrier frequency offset and makes CFO compensation in thefrequency domain computationally inefficient. Accordingly, in someembodiments CFO is corrected in the time domain prior to the conversionof the received signal into the frequency domain (e.g., in phase one ofmethod 300, FIG. 3B).

In some embodiments, correction of the carrier frequency offset involvescorrection of the (estimated) receiver IQ offset (e.g., as illustratedin methods 350 and 300, FIGS. 3A and 3B). For example, method 350 or 300(FIGS. 3A and 3B) is used to correct the carrier frequency offset andreceiver IQ mismatch if a carrier frequency offset is present.Frequency-domain estimation and correction are then used to compensatefor the influence of the channel, transmitter IQ offset, and anyresidual receiver IQ offset. If no carrier frequency offset is present,a joint estimate of the channel, transmitter IQ offset, and receiver IQoffset is obtained.

FIG. 7 illustrates successive pairs of OFDM symbols transmitted by atransmitter 102 (FIG. 1A) and received by a receiver 106 (FIG. 1A). TheOFDM symbols include known pilot symbols 702 on different sets ofsubcarriers (the placing of the pilot symbols 702 is indicated by theboxes in FIG. 7). These pilot symbols 702 are known modulation symbolsplaced on respective subcarriers. The OFDM symbols are indexed by asymbol index (“symbol idx”) and the subcarriers are indexed by asubcarrier index (“subcarrier idx”). Successive pairs of OFDM symbolscompose respective subframes, which are indexed by a subframe index(“subframe idx”). For example, OFDM symbols 0 and 1 compose subframe 0,OFDM symbols 2 and 3 compose subframe 1, and so on. A specified numberof subframes (e.g., 16 subframes) compose a frame. Frames are indexed bya frame index (“frame idx”). FIG. 7 shows only a snapshot (i.e., aportion) of the available subcarriers. For example, there may be 4096,8192, or 16,384 subcarriers.

Pilot symbols 702 are placed on the same subcarriers in both OFDMsymbols of a subframe (e.g., in accordance with Equation (6) for pilotmatrices) and are symmetric (e.g., mirrored) about a center carrierfrequency, which is the DC subcarrier in baseband. The subcarriers usedfor pilot symbols 702 are evenly spaced on each side of the centercarrier frequency. For example, 1 of every 64 subcarriers is used for apilot symbol 702. For OFDM symbols 0 and 1 (i.e., subframe 0),subcarriers 2, 66, 130, and so on, and also subcarriers −2, −66, −130,and so on, are used for pilot symbols 702. For OFDM symbols 2 and 3(i.e., subframe 1), subcarriers 34, 98, 162, and so on, and alsosubcarriers −34, −98, −162, and so on, are used for pilot symbols 702.The pilot symbol overhead is thus 1/64 (or more generally, a predefinedfraction), resulting in a corresponding reduction in spectralefficiency; the remaining 63 of each 64 subcarriers may be used for datatransmission.

Within a given subframe, the pilot symbols 702 are generated inaccordance with equation (6) or a variant of equation (6) in which oneof the columns of the orthogonal pilot matrix of equation (6) ismultiplied by −1 (thus maintaining the orthogonality). In the pilotmatrix of equation (6), or variants thereof, the first columncorresponds to the first OFDM symbol in the subframe and the secondcolumn corresponds to the second OFDM symbol in the subframe. The firsttwo entries in each column correspond to the real and imaginarycomponents of a pilot symbol 702 above the center carrier frequency andthe second two entries in each column correspond to the real andimaginary components of a pilot symbol 702 below the center carrierfrequency. For example, the first OFDM symbol includes a first pilotsymbol 702 on a subcarrier above the center carrier frequency (i.e., apositive subcarrier) and a second pilot symbol 702 on a subcarrier belowthe center carrier frequency (i.e., a negative subcarrier), and thesecond OFDM symbol includes the first pilot symbol 702 on the subcarrierbelow the center carrier frequency and the negative of the second pilotsymbol 702 on the subcarrier above the center carrier frequency.Alternatively, the second OFDM symbol includes the second pilot symbol702 on the subcarrier above the center carrier frequency and thenegative of the first pilot symbol 702 on the subcarrier below thecenter carrier frequency. In these examples, the subcarrier above thecenter carrier frequency and the subcarrier below the center carrierfrequency (i.e., the negative and positive subcarriers) are symmetricabout the center carrier frequency.

The pilot symbol subcarriers in the different subframes of a frame arestaggered with respect to each other, such that a predefined fraction ofsubcarriers are used for pilot symbols 702 somewhere in the frame. Thesubcarriers to be used for pilot symbols 702 in a given subframe may bedetermined by averaging the indices of the subcarriers used in twoprevious subframes, resulting in a pattern that is staggered in time asshown in FIG. 7. In the example of FIG. 7, every fourth subcarrier isused for pilot symbols 702 in some subframe within the frame. Each framein this example thus includes OFDM symbols with 16 different sets ofpilot positions.

For each subframe, the receiver 106 (FIG. 1A) estimates frequencyresponses for the subcarriers of the pilot symbols 702 (e.g., usingEquation (7)) and compensates accordingly for the estimated frequencyresponse, which represents signal impairment. Thus, in the example ofFIG. 7, frequency response estimation is performed for every fourthsubcarrier during reception of each frame. Frequency responses for theremaining subcarriers (i.e., the subcarriers not used for pilot symbols702 anywhere in the frame) may be interpolated and compensated foraccordingly.

Including pilot symbols 702 in each OFDM symbol (e.g., staggered in apattern such as the pattern of FIG. 7, or alternatively in the samesubcarriers from OFDM symbol to OFDM symbol) enables continuous trackingof phase noise and carrier frequency offset, while using only a smallamount of overhead (e.g., 1/64). Furthermore, the symmetry of the pilotsymbol subcarriers about the center carrier frequency and the use of thesame subcarriers for pilot symbols 702 in a pair of OFDM symbols (e.g.,in accordance with Equation (6)) enables correction of transmitterand/or receiver IQ mismatch (e.g., in accordance with Equation (7)).

The spacing of subcarriers used for pilot symbols 702 within a frame maybe determined based on the minimal coherence bandwidth, whichcorresponds to a maximal delay spread equal to the cyclic prefix length.For example, for a cyclic prefix length of 4 us, and thus a maximaldelay spread of 4 us, the minimal coherence bandwidth is 250 kHz. Thespacing of pilot symbol subcarriers may be a specified fraction of thecoherence bandwidth. For example, if the spacing between subcarriers inFIG. 7 is 12.5 kHz, the spacing between subcarriers used for pilotsymbols 702 somewhere within a frame is 50 kHz, or one-fifth of thecoherence bandwidth, since every fourth subcarrier is used for pilotsymbols 702 at some point within a frame.

Attention is now directed to joint estimation of and correction for bothtransmitter-side I/Q mismatch and receiver-side I/Q mismatch. In someembodiments, carrier frequency offset is estimated, after which jointestimation of transmitter-side I/Q mismatch and receiver-side I/Qmismatch is performed.

Let us define p₁=a_(Rx)a_(Tx), p₂=a_(Rx)b_(Tx), p₃=b_(Rx)a_(Tx)*,p₄=b_(Rx)b_(Tx)*. These four unknown parameters can be estimated in asimilar fashion as in the case of receiver-only IQ mismatch. Thereceived signal vector z[n] can be written as

${z\lbrack n\rbrack} = {{\begin{bmatrix}{{\overset{\sim}{x}}_{1}\lbrack n\rbrack} & {{\overset{\sim}{x}}_{2}\lbrack n\rbrack} & {{\overset{\sim}{x}}_{3}\lbrack n\rbrack} & {{\overset{\sim}{x}}_{4}\lbrack n\rbrack}\end{bmatrix}\begin{bmatrix}p_{1} \\p_{2} \\p_{3} \\p_{4}\end{bmatrix}} + {\overset{\sim}{w}\lbrack n\rbrack}}$

where x₁[n]=e^(jΔωn)x[n], {tilde over (x)}₂[n]=e^(jΔωn)x*[n], {tildeover (x)}₃[n]=e^(−jΔωn)x*[n], {tilde over (x)}₄[n]=e^(−jΔωn)x[n].Inversion of the N×4 system matrix to solve for the four unknownparameters can be avoided by employing adaptive filtering techniquessuch as LMS.

Once the signal parameters have been estimated, CFO, transmitter-sideI/Q mismatch, and receiver-side I/Q mismatch can be compensated. Thereceived signal can be re-written as

z[n]=(p ₁ e ^(jΔωn) +p ₄ e ^(−jΔωn))x[n]+(p ₂ e ^(jΔωn) +p ₃ e^(−jΔωn))x*[n]+a _(Rx) w[n]+b _(Rx) w*[n]

Transmitter-side I/Q mismatch and receiver-side I/Q mismatch can becompensated jointly via the following transformation:

${\hat{z}\lbrack n\rbrack} = {\frac{1}{{{u\lbrack n\rbrack}}^{2} - {{s\lbrack n\rbrack}}^{2}}\left\lbrack {{{u^{*}\lbrack n\rbrack}{z\lbrack n\rbrack}} - {{s\lbrack n\rbrack}{z^{*}\lbrack n\rbrack}}} \right\rbrack}$

where u[n]=p₁e^(jΔωn)+p₄e^(−jΔωn) and s[n]=p₂e^(jΔωn)+p₃e^(−jΔωn). Thesignal after joint CFO and I/Q compensation reads

{circumflex over (z)}[n]=x[n]+{tilde over (w)}[n]

For small I/Q mismatch, such that we can approximate f[n]≈1, sin Δφ≈Δφ,cos Δφ≈1, g[n] sin Δφ≈0, the signal model simplifies to

z[n]=e ^(jΔωn) x[n]+b _(Tx) e ^(jΔωn) x*[n]+b _(Rx) e ^(−jΔωn)x*[n]+w[n]+b _(Rx) w*[n]

This simplification allows the transmitter-side I/Q mismatch andreceiver-side I/Q mismatch to be estimated separately. In someembodiments, the correction procedure would first compensate forreceiver-side I/Q mismatch, then compensate for the CFO (e.g., via phaserotation), and then compensate for the transmitter-side I/Q mismatch.For example, the method 300 (FIG. 3B) would be used for the correctionprocedure.

In some embodiments the method 350 (FIG. 3A) is used for jointcompensation, wherein operations 354 and 360 estimate the jointreceiver-side and transmitter-side I/Q mismatch and operations 356 and362 compensate for the joint I/Q mismatch.

Joint estimation of and correction for both transmitter-side I/Qmismatch and receiver-side I/Q mismatch may be performed for widebandsignals (e.g., after the carrier frequency offset has been estimated).In some embodiments, if signals occupy a wide bandwidth, multipathchannel effects are considered, as is the frequency dependency of I/Qmismatch at both the transmitter side and receiver side. FIG. 2Eillustrates signal impairments for wideband signals, including a channeltransfer function h[n] for a channel 130.

The overall received signal at the receiver can be expressed as the sumof four signal components and an effective noise component

z[n]=a _(Rx) [n]*[e ^(jΔωn)(h[n]*a _(Tx) [n]*x[n])]+a _(Rx) [n]*[e^(jΔωn)(h[n]*b _(Tx) [n]*x*[n])]+b _(Rx) [n]*[e ^(−jΔωn)(h*[n]*a _(Tx)*[n]*x*[n])]+b _(Rx) [n]*[e ^(−jΔωn)(h*[n]*b _(Tx) *[n]*x[n])]+a _(Rx)[n]*w[n]+b _(Rx) [n]*w*[n]

The signal components other than the first constitute the interferencedue to transmitter I/Q mismatch and receiver I/Q mismatch. The lastinterference term is presumably quite weak as compared to the precedingtwo.

Before compensating for I/Q mismatch, the CFO is estimated by employinga narrowband signal (e.g., a narrowband training signal). In someembodiments, the I/Q mismatch is then estimated separately for thereceiver side and the transceiver side, by analogy for the narrowbandcase, using convolution matrices to account for the frequency dependencyof the I/Q mismatch.

Alternatively, joint estimation and correction of transmitter-side I/Qmismatch and receiver-side I/Q mismatch is performed. If the CFO issmall compared to the coherence bandwidth of the channel andtransmitter/receiver I/Q filters, the frequency shift due to CFO can becorrected before filtering operations. Therefore, the received signal inthe frequency domain can be expressed as

${Z(f)} = {{{P_{1}(f)}{X\left( {f + \frac{\Delta \; \omega}{2\pi}} \right)}} + {{P_{2}(f)}{X^{*}\left( {{- f} + \frac{\Delta \; \omega}{2\pi}} \right)}} + {{P_{3}(f)}{X^{*}\left( {{- f} - \frac{\Delta \; \omega}{2\pi}} \right)}} + {{P_{4}(f)}{X\left( {f - \frac{\Delta \; \omega}{2\pi}} \right)}} + {\overset{\sim}{W}(f)}}$

Estimation and compensation are performed using the method describedabove with respect to FIG. 6 that results in equation (7), in accordancewith some embodiments.

Attention is now directed to a method of communication using pilotsymbols, such as the pilot symbols 702 (FIG. 7). FIG. 8 is a flowchartshowing a method 800 of communicating between an OFDM transmitter (e.g.,the transmitter 102, FIG. 1A) and an OFDM receiver (e.g., the receiver106, FIG. 1A) in accordance with some embodiments.

The OFDM transmitter transmits (802) successive pairs of OFDM symbols(e.g., successive subframes that include pilot symbols 702 on both OFDMsymbols of each subframe, as shown in FIG. 7). Both OFDM symbols of arespective pair include pilot symbols on one or more groups of twosubcarriers. The two subcarriers of each group are the same for bothOFDM symbols and are symmetric about a center carrier frequency. Thepilot symbols on the two subcarriers compose an orthogonal matrix (e.g.,in accordance with equation (6) or a variant of equation (6) in whichone of the columns is multiplied by −1).

In some embodiments, the OFDM symbols of the successive pairs include(804) pilot symbols on respective subsets of a plurality of subcarriers(e.g., as shown in FIG. 7). Both OFDM symbols of each of the successivepairs have pilot symbols on a respective subset of the plurality ofsubcarriers. The respective subsets of the plurality of subcarriers aredistinct.

In some embodiments, the respective subsets of the plurality ofsubcarriers include (805) subcarriers that are evenly spaced on eachside of a center carrier frequency and have mirror symmetry about thecenter carrier frequency. For example, the pilot symbols 702 have mirrorsymmetry about the DC subcarrier and may be placed on a predefinedfraction of subcarriers in an evenly spaced manner (e.g., on every 64thsubcarrier in an OFDM symbol).

In some embodiments, the respective subsets of the plurality ofsubcarriers are staggered (806) with respect to each other. For example,the index of a subcarrier on which pilot symbols are placed for a pairof OFDM symbols may be determined by averaging the indices ofsubcarriers on which pilot symbols were placed for two previous pairs ofsubcarriers. In FIG. 7, the pair of OFDM symbols in subframe 0 includespilot symbols on subcarriers ±2, the pair of OFDM symbols in subframe 1includes pilot symbols on subcarriers ±34, and the pair of OFDM symbolsin subframe 2 includes pilot symbols on subcarriers ±18. The subcarrierindices for the pilot symbols in subframe 2 are determined by averagingthe subcarrier indices for the pilot symbols in subframes 0 and 1: 18 isthe average of 2 and 34, and −18 is the average of −2 and −34.Similarly, the pair of OFDM symbols in subframe 4 includes pilot symbolson subcarriers ±10, as determined by averaging subcarrier indices forsubframes 0 and 2.

In some embodiments, pilot symbols for respective pairs (e.g., eachpair) of OFDM symbols are generated in accordance with equation (6) or avariant of equation (6) in which one of the columns of the orthogonalpilot matrix of equation (6) is multiplied by −1 (thus maintaining theorthogonality). For example, the first OFDM symbol of a pair includes afirst pilot symbol on a first subcarrier above a center carrierfrequency and a second pilot symbol on a second subcarrier below thecenter carrier frequency and symmetric with the first subcarrier aboutthe center carrier frequency. The second OFDM symbol of the pairincludes the first pilot symbol on the second subcarrier and thenegative of the second pilot symbol on the first subcarrier.Alternatively, the second OFDM symbol of the pair includes the negativeof the first pilot symbol on the second subcarrier and the second pilotsymbol on the first subcarrier.

The OFDM receiver receives (808) the successive pairs of OFDM symbols.Using the pilot symbols, the OFDM receiver estimates (810) frequencyresponses at frequencies corresponding to the subcarriers carrying pilotsymbols (e.g., to the respective subsets of the plurality ofsubcarriers). This estimation is performed, for example, using equation(7). In some embodiments, the OFDM receiver interpolates (812) frequencyresponses for subcarriers not carrying pilot symbols (e.g., forsubcarriers not included in the respective subsets of the plurality ofsubcarriers), based on the estimated frequency responses. The OFDMreceiver compensates (814) for signal impairment based at least in parton the estimated frequency responses. In some embodiments, the OFDMreceiver compensates (816) for the signal impairment based further onthe interpolated frequency responses.

While the method 800 includes a number of operations that appear tooccur in a specific order, it should be apparent that the method 800 caninclude more or fewer operations, which can be executed serially or inparallel. Performance of two or more operations may overlap and two ormore operations may be combined into a single operation. For example,all of the operations of the method 800 may be performed in an ongoingbasis.

FIG. 5 is an example of a block diagram of a communication device 500that performs signal impairment estimation and compensation. In someembodiments, the device 500 is a wireless device (e.g., a WLAN device,such as a personal computer, laptop or tablet computer, mobile phone,personal digital assistant, GPS device, wireless access point, or otherelectronic device). In some embodiments, the device 500 has a wirednetwork connection.

The device 500 includes a processor unit 501, memory unit 507, networkinterface 505, and transceiver 200 (FIG. 2A) coupled by a bus 503. Theprocessor unit 501 includes one or more processors and/or processorcores. In some embodiments, the network interface 505 includes at leastone wireless network interface (e.g., a WLAN interface, a Bluetooth®interface, a WiMAX interface, a ZigBee® interface, a Wireless USBinterface, etc.). In some embodiments, the device 500 includes at leastone wired network interface (e.g., to interface with a coaxial cable orother physical medium).

The memory unit 507 includes a non-transitory computer-readable storagemedium (e.g., one or more nonvolatile memory elements, such as EPROM,EEPROM, Flash memory, a hard disk drive, and so on) that stores a signalimpairment estimation and compensation software module 510. In someembodiments, the software module 510 includes one or more programs withinstructions that, when executed by processor unit 501 and/or by thereceiver baseband processor 280 (FIG. 2A), cause the mobile device 500to perform the methods 300 and/or 350 (FIGS. 3A-3B). In someembodiments, these instructions include instructions for performingtime-domain compensation (e.g., as described with regard to FIGS. 4A-4Dand equations 1-5) and/or frequency domain compensation. In someembodiments, these instructions include instructions for separatelyestimating and compensating for transmitter IQ mismatch and receiver IQmismatch and/or for jointly estimating and compensating for transmitterIQ mismatch and receiver IQ mismatch, using any technique describedherein. In some embodiments, these instructions include instructions forperforming all or part of the transmitter-side and/or receiver-sideportions of the method 800 (FIG. 8).

In the foregoing specification, the present embodiments have beendescribed with reference to specific exemplary embodiments thereof. Itwill, however, be evident that various modifications and changes may bemade thereto without departing from the broader spirit and scope of thedisclosure as set forth in the appended claims. The specification anddrawings are, accordingly, to be regarded in an illustrative senserather than a restrictive sense.

What is claimed is:
 1. A method of operating an orthogonalfrequency-division multiplexing (OFDM) receiver, comprising: receiving apair of OFDM symbols, wherein first and second OFDM symbols of the pairboth comprise pilot symbols on two subcarriers that are symmetric abouta center carrier frequency, the two subcarriers are the same for thefirst and second OFDM symbols, and the pilot symbols on the twosubcarriers for the first and second OFDM symbols compose an orthogonalmatrix; estimating frequency responses at frequencies of the twosubcarriers; and compensating for signal impairment based at least inpart on the estimated frequency responses at the frequencies of the twosubcarriers.
 2. The method of claim 1, wherein: the two subcarrierscomprise a first subcarrier above the center carrier frequency and asecond subcarrier below the center carrier frequency; the first OFDMsymbol comprises a first pilot symbol on the first subcarrier and asecond pilot symbol on the second subcarrier; and the second OFDM symbolcomprises the first pilot symbol on the second subcarrier and thenegative of the second pilot symbol on the first subcarrier.
 3. Themethod of claim 1, wherein: the two subcarriers comprise a firstsubcarrier above the center carrier frequency and a second subcarrierbelow the center carrier frequency; the first OFDM symbol comprises afirst pilot symbol on the first subcarrier and a second pilot symbol onthe second subcarrier; and the second OFDM symbol comprises the negativeof the first pilot symbol on the second subcarrier and the second pilotsymbol on the first subcarrier.
 4. The method of claim 1, wherein thepair of OFDM symbols is a first pair, the method further comprising:receiving successive pairs of OFDM symbols including the first pair, theOFDM symbols of the successive pairs comprising pilot symbols ondistinct respective subsets of a plurality of subcarriers, both OFDMsymbols of each of the successive pairs having pilot symbols on arespective subset of the plurality of subcarriers, the respectivesubsets being symmetric about the center carrier frequency; estimatingfrequency responses at frequencies of the respective subsets of theplurality of subcarriers; and compensating for signal impairment basedat least in part on the estimated frequency responses at the frequenciesof the respective subsets of the plurality of subcarriers.
 5. The methodof claim 4, wherein the respective subsets of the plurality ofsubcarriers are staggered with respect to each other.
 6. The method ofclaim 5, wherein: the successive pairs of OFDM symbols comprise thefirst pair, a second pair subsequent to the first pair, and a third pairsubsequent to the second pair; the first pair comprises pilot symbols ona subcarrier having a first index; the second pair comprises pilotsymbols on a subcarrier having a second index; and the third paircomprises pilot symbols on a subcarrier having a third index equal to anaverage of the first index and the second index.
 7. The method of claim4, wherein: the successive pairs of OFDM symbols compose a frame; andthe frame includes pilot symbols on a specified fraction of theplurality of subcarriers.
 8. The method of claim 4, further comprisinginterpolating frequency responses for subcarriers not included in therespective subsets of the plurality of subcarriers, based on theestimated frequency responses at the frequencies of the respectivesubsets of the plurality of subcarriers; wherein compensating for thesignal impairment is further based on the interpolated frequencyresponses.
 9. A method of operating an orthogonal frequency-divisionmultiplexing (OFDM) transmitter, comprising: transmitting a pair of OFDMsymbols, wherein first and second OFDM symbols of the pair both comprisepilot symbols on two subcarriers that are symmetric about a centercarrier frequency, the two subcarriers are the same for the first andsecond OFDM symbols, and the pilot symbols on the two subcarriers forthe first and second OFDM symbols compose an orthogonal matrix.
 10. Themethod of claim 9, wherein: the two subcarriers comprise a firstsubcarrier above the center carrier frequency and a second subcarrierbelow the center carrier frequency; the first OFDM symbol comprises afirst pilot symbol on the first subcarrier and a second pilot symbol onthe second subcarrier; and the second OFDM symbol comprises the firstpilot symbol on the second subcarrier and the negative of the secondpilot symbol on the first subcarrier.
 11. The method of claim 9,wherein: the two subcarriers comprise a first subcarrier above thecenter carrier frequency and a second subcarrier below the centercarrier frequency; the first OFDM symbol comprises a first pilot symbolon the first subcarrier and a second pilot symbol on the secondsubcarrier; and the second OFDM symbol comprises the negative of thefirst pilot symbol on the second subcarrier and the second pilot symbolon the first subcarrier.
 12. The method of claim 9, wherein the pair ofOFDM symbols is a first pair, the method further comprising transmittingsuccessive pairs of OFDM symbols including the first pair, wherein theOFDM symbols of the successive pairs comprises pilot symbols on distinctrespective subsets of a plurality of subcarriers, both OFDM symbols ofeach of the successive pairs include pilot symbols on a respectivesubset of the plurality of subcarriers, and the respective subsets aresymmetric about the center carrier frequency.
 13. The method of claim12, wherein the respective subsets of the plurality of subcarriers arestaggered with respect to each other.
 14. The method of claim 13,wherein transmitting the successive pairs of OFDM symbols comprises: inthe first pair, placing pilot symbols on a subcarrier having a firstindex; in a second pair subsequent to the first pair, placing pilotsymbols on a subcarrier having a second index; and in a third pairsubsequent to the first and second pairs, placing pilot symbols on asubcarrier having a third index equal to an average of the first indexand the second index.
 15. The method of claim 12, wherein: thesuccessive pairs of OFDM symbols compose a frame; and the frame includespilot symbols on a specified fraction of the plurality of subcarriers.16. A communications device, comprising an orthogonal frequency-divisionmultiplexing (OFDM) receiver to: receive successive pairs of OFDMsymbols including a first pair, wherein first and second OFDM symbols ofthe first pair both comprise pilot symbols on two subcarriers that aresymmetric about a center carrier frequency, the two subcarriers are thesame for the first and second OFDM symbols, and the pilot symbols on thetwo subcarriers for the first and second OFDM symbols compose anorthogonal matrix; estimate frequency responses at frequencies includingthe frequencies of the two subcarriers; and compensate for signalimpairment based at least in part on the estimated frequency responses.17. The communications device of claim 16, wherein: the two subcarrierscomprise a first subcarrier above the center carrier frequency and asecond subcarrier below the center carrier frequency; the first OFDMsymbol of the first pair comprises a first pilot symbol on the firstsubcarrier and a second pilot symbol on the second subcarrier; and thesecond OFDM symbol of the first pair comprises the first pilot symbol onthe second subcarrier and the negative of the second pilot symbol on thefirst subcarrier.
 18. The communications device of claim 16, wherein:the two subcarriers comprise a first subcarrier above the center carrierfrequency and a second subcarrier below the center carrier frequency;the first OFDM symbol of the first pair comprises a first pilot symbolon the first subcarrier and a second pilot symbol on the secondsubcarrier; and the second OFDM symbol of the first pair comprises thenegative of the first pilot symbol on the second subcarrier and thesecond pilot symbol on the first subcarrier.
 19. The communicationsdevice of claim 16, wherein: the OFDM symbols of the successive pairscomprise pilot symbols on distinct respective subsets of a plurality ofsubcarriers, both OFDM symbols of each of the successive pairs includepilot symbols on a respective subset of the plurality of subcarriers,and the respective subsets are symmetric about the center carrierfrequency; and the OFDM receiver is to estimate frequency responses atfrequencies of the respective subsets of the plurality of subcarriers.20. The communications device of claim 19, wherein the respectivesubsets of the plurality of subcarriers are staggered with respect toeach other.
 21. The communications device of claim 20, wherein: thesuccessive pairs of OFDM symbols comprise the first pair, a second pairsubsequent to the first pair, and a third pair subsequent to the secondpair; the first pair comprises pilot symbols on a subcarrier having afirst index; the second pair comprises pilot symbols on a subcarrierhaving a second index; and the third pair comprises pilot symbols on asubcarrier having a third index equal to an average of the first indexand the second index.
 22. The communications device of claim 19,wherein: the successive pairs of OFDM symbols compose a frame; and theframe includes pilot symbols on a specified fraction of the plurality ofsubcarriers.
 23. The communications device of claim 19, wherein: theOFDM receiver is further to interpolate frequency responses forsubcarriers not included in the respective subsets of the plurality ofsubcarriers, based on the estimated frequency responses at thefrequencies corresponding to the respective subsets of the plurality ofsubcarriers; and the OFDM receiver is to compensate for the signalimpairment based further on the interpolated frequency responses.
 24. Acommunications device, comprising an orthogonal frequency-divisionmultiplexing (OFDM) transmitter to transmit successive pairs of OFDMsymbols including a first pair, wherein: first and second OFDM symbolsof the first pair both comprise pilot symbols on two subcarriers thatare symmetric about a center carrier frequency; the two subcarriers arethe same for the first and second OFDM symbols; and the pilot symbols onthe two subcarriers for the first and second OFDM symbols compose anorthogonal matrix.
 25. The communications device of claim 24, wherein:the two subcarriers comprise a first subcarrier above the center carrierfrequency and a second subcarrier below the center carrier frequency;the first OFDM symbol of the first pair comprises a first pilot symbolon the first subcarrier and a second pilot symbol on the secondsubcarrier; and the second OFDM symbol of the first pair comprises thefirst pilot symbol on the second subcarrier and the negative of thesecond pilot symbol on the first subcarrier.
 26. The communicationsdevice of claim 24, wherein: the two subcarriers comprise a firstsubcarrier above the center carrier frequency and a second subcarrierbelow the center carrier frequency; the first OFDM symbol of the firstpair comprises a first pilot symbol on the first subcarrier and a secondpilot symbol on the second subcarrier; and the second OFDM symbol of thefirst pair comprises the negative of the first pilot symbol on thesecond subcarrier and the second pilot symbol on the first subcarrier.27. The communications device of claim 24, wherein the OFDM symbols ofthe successive pairs comprise pilot symbols on distinct respectivesubsets of a plurality of subcarriers, both OFDM symbols of each of thesuccessive pairs include pilot symbols on a respective subset of theplurality of subcarriers, and the respective subsets are symmetric aboutthe center carrier frequency.
 28. The communications device of claim 27,wherein the respective subsets of the plurality of subcarriers arestaggered with respect to each other.
 29. The communications device ofclaim 28, wherein the transmitter is to: place pilot symbols on asubcarrier having a first index in the first pair of OFDM symbols; placepilot symbols on a subcarrier having a second index in a second pair ofOFDM symbols subsequent to the first pair; and place pilot symbols on asubcarrier having a third index equal to an average of the first indexand the second index in a third pair of OFDM symbols subsequent to thefirst and second pairs.
 30. The communications device of claim 27,wherein: the successive pairs of OFDM symbols compose a frame; and theframe includes pilot symbols on a specified fraction of the plurality ofsubcarriers.